14 Wave-particle nonlinearities
14.1 Introduction
Linear models are straightforward and rich in descriptive power becausethey are based
on a powerful, integrated set of concepts and tools consisting of eigenmodes, eigenvalues,
eigenvectors, orthogonality, and integral transforms. These concepts and tools ultimately
depend on the very essence of linearity, namely the principle of superposition.
While many important phenomena are well characterized by linear models there never-
theless exist other important phenomena where the principle of superposition breaks down
either partially or completely. The phenomenon in question becomes amplitude-dependent
above some critical amplitude threshold and then nonlinearity becomes important. Break-
down of the superposition principle means that modes with different eigenvalues (to the
extent eigenvalues still exist) are no longer orthogonal and start to interact with each other.
These interactions result from non-linear terms in the system of equations, i.e., terms in-
volving products of dependent variables. As an example of this, consider the product of
two modes having respective frequenciesω 1 andω 2 .This product can be decomposed into
sum and difference frequencies according to standard trigonometric identities, e.g.,
cosω 1 tcosω 2 t=cos[(ω 1 +ω 2 )t]+cos[(ω 1 −ω 2 )t]
2. (14.1)
The beat waves at frequenciesω 1 ±ω 2 can act as source terms driving oscillations at
ω 1 ±ω 2. In the special case whereω 1 =ω 2 , the difference frequency is zero and the non-
linear product can act as a source term which modulates equilibrium parameters and thereby
changes the mode dynamics. In particular, feedback loops can develop where modes affect
their own stability properties. Another possibility occurs when the nonlinear product is very
small, but happens to resonantly drive a linear mode. In this case even a weak non-linear
coupling between two modes can resonantly drive another linear mode to large amplitude.
Similar beating can occur with spatial factors∼exp(ik·x)so that the non-linear product
of modes with wavevectorsk 1 andk 2 drive spatial oscillations with wavevectorsk 1 ±k 2.
In analogy to quantum mechanics the momentum associated with a wave is foundto be
proportional tokand the energy proportional toω.The beating together of two waves with
respective space-time dependenceexp(ik 1 ·x−iω 1 t)andexp(ik 2 ·x−iω 2 t)can then be
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